Question: Determine how many solutions exist for the system of equations. ${-3x-3y = -15}$ ${x+y = 5}$
Explanation: Convert both equations to slope-intercept form: ${-3x-3y = -15}$ $-3x{+3x} - 3y = -15{+3x}$ $-3y = -15+3x$ $y = 5-x$ ${y = -x+5}$ ${x+y = 5}$ $x{-x} + y = 5{-x}$ $y = 5-x$ ${y = -x+5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -x+5}$ ${y = -x+5}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-3x-3y = -15}$ is also a solution of ${x+y = 5}$, there are infinitely many solutions.